A lot depends on how fat the pawns are as well. Sorry, that probably didn't help... You could always take a ruler and draw a paper chessboard and see how you like it.
King base vs. square size
Hi, have a look here: http://www.regencychess.co.uk/size_guide.html Of course you can always have you own preference. 😀

Nathan,
A 1.75 diameter king is not necessarily too large for a 2.16" chess board (81%), but there are other factors to consider besides the diameter of the king, such as the diameter of the pawns, which just happen to be 50% of all the pieces on the board. Whereas the two kings represent just over 6% of all the pieces.
So it makes little sense to right-size two pieces if, in doing so, you're left with sixteen pieces that are "wrong-sized." In addition, the dimensions of the remaining pieces can also factor in. You may get by with a set where the two kings fit a bit snugly, but what if the queens are nearly the same diameter as the king? Now you've got four oversized pieces on a board where everything else seems to fit just right. I once had a set with oversized knights where I had to move up a size (1/8") in order for the set to look balanced. The bottom line is, of course, how things look to you.
Here's a link to a system I devised for sizing chess pieces and boards.
https://www.chess.com/forum/view/chess-equipment/a-system-for-sizing-chess-pieces-and-boards-long
P.S. I use these formulas not just to match chess sets and boards, but, as importantly, to make sure the chess sets I'm considering are proportionally balanced -- that is, where the kings, pawns, and remaining pieces are properly scaled to fit a specific size chess board. Once I've determined the ideal board, I can always move up or down to the next size board depending on my preference or necessity.

The primary sizing criteria is that of square size vs King base diameter...
A good rule of thumb is...
King base diameter = 0.75 * square size
or equivalently....
Square size = 1.33 * King base diameter
Any variation from these ratios, or the size of the pieces and pawns is, of course, a matter of personal preference...
A lot depends on how fat the pawns are as well. Sorry, that probably didn't help... You could always take a ruler and draw a paper chessboard and see how you like it.
That's a good idea! I tried it (but with only one square), and it seems ok. Still thinking about it.
Hi, have a look here: http://www.regencychess.co.uk/size_guide.html Of course you can always have you own preference. 😀
Thanks very much for that. Inspired by the Regency link, I looked for FIDE recommendations, and found that in their handbook, they say that the king should be about 3.74", with a base of 40% to 50% of its height--but then they give an extra 10% of wriggle room. In relation to the squares, they actually mention the pawn, rather than the king. They say that four pawns should be able to fit on one square.
In my case, I can fit most of four pawns on one 2.16" square--they jut out just slightly on all four sides.
Thank you, loubalch. I found your very interesting post after I had posted my question, and I was hoping you'd weigh in! It's very late, and I can't go into too much detail at the moment in terms of the more technical aspects of things. But can you take a look at this board set up:
Now this would obviously not work for a tournament setting, but do you think it is too cramped for informal play? Now the good news is: I don't really intend to pair these pieces up with this board. The pieces I'd like to pair up with this board aren't in my possession, yet, but they are smaller. The king base of that set is the 1.625"; the board squres are 2", so it would be, again, 81%.
The next photo shows the same pieces (HOS Players) on the 2.25" board. I confess that I like them on that board more...
...but I can't help but wondering if I could fit these pieces on to a board with 2.16" squares (81% for the king, again).
I don't own either the board with the 2.16" squares, or the pieces with the king base of 1.625" diameter. I can't get them both now, either; but they are possible options for the future.
The primary sizing criteria is that of square size vs King base diameter...
A good rule of thumb is...
King base diameter = 0.75 * square size
or equivalently....
Square size = 1.33 * King base diameter
Any variation from these ratios, or the size of the pieces and pawns is, of course, a matter of personal preference...
Thanks, Russ; so I guess I'm wondering: is 81% (rather than our ideal 75%) too outside of the realm of what is reasonable or tasteful for most people?
While I'm on the subject: why is is to very hard to find a nice set at a reasonable price that is intended for use on a board with 2" squares?

The primary sizing criteria is that of square size vs King base diameter...
A good rule of thumb is...
King base diameter = 0.75 * square size
or equivalently....
Square size = 1.33 * King base diameter
Any variation from these ratios, or the size of the pieces and pawns is, of course, a matter of personal preference...
Thanks, Russ; so I guess I'm wondering: is 81% (rather than our ideal 75%) too outside of the realm of what is reasonable or tasteful for most people?
IMO 81% is about the upper limit for a reasonable ratio. I would not exceed it. It is also a ratio which, for example, ChessHouse.com appears to imply as an upper limit. (e.g., see next answer).
While I'm on the subject: why is is to very hard to find a nice set at a reasonable price that is intended for use on a board with 2" squares?
Here is what I consider to be an example of a nice wood set, not too expensive, with a King height of 3.75 inches and King base of 1.625 inches, which puts the ratio at 81% when used with a board with 2 inch squares (1.625/2.0 = .8125). Note that the ChessHouse description of the pieces indicates that, in their opinion, the pieces would work with a board with 2 inch squares or larger. By the way, I can vouch that ChessHouse is a reputable retailer to do business with; the owner, Raphael Neff, cares about the quality of service he provides and has a good reputation. I just noticed that on the website it indicates the pieces are not available at the moment. You can contact them to find out if/when they will become available. They will give you an honest answer.
https://www.chesshouse.com/collections/wooden-chess-pieces/products/3-3-4-executive-staunton-ebonized-wood-chess-pieces
Here is another relatively inexpensive wood set, which to me looks fine, from WholesaleChess.com, with the same dimensions....check the photo of the pieces on the board, to get a sense of that they really look like - appears to be the rosewood King & Rook, and the ebonized pawn & Bishop (hover mouse over photo to magnify)....
https://www.wholesalechess.com/shop/chess-pieces/wood-chess-pieces/supreme-wood-chess-pieces-3-75-king

Nathan,
Thanks for the pictures. You've given us the scaling factor for the king on both boards, but what's the scaling factor of the pawns in photos above?
That's my whole point, when matching chess pieces and boards if your only consideration is the dimensions of the king, you're not getting the full picture.
Yes, in the first picture above the pieces fit a bit snugly, but not overly so. I have the Ultimate wooden set with a 1.66" diameter king and 1.18" diameters pawns, which, on a 2" board, gives me a combined scaling factor of 83% and 58.8%. I've been using this setup for the last few weeks at the chess club and at other outings and no one's complained about overcrowding.
On the 2.25" board above I think the back row pieces look fine, but the pawns look whimpy, which makes the whole set look somewhat dis-proportioned. I'm not a fan of scrawny "4 pawns to a square" scaling (50%). I much prefer stouter pawns with a scaling factor between 55-58%. To my eye, it makes everything look more balanced,
For me, your set would work best on a 2-1/8" board. And there's my beef, there are plenty of chess sets out there with similar proportions, but where are the 2" and 2.125" vinyl and mousepad chess boards? Zip, nada, nowhere.
Why? Because players have accepted 1.5" diameter kings on 2.25" chessboards (your typical cheap plastic set with vinyl roll up board) as standard fare at chess clubs and tournaments.
An example of what I consider a well-balanced set is the following 3.75" set, with a 1.5" diameter king and 1.13" diameter pawns and a 2" chess board, for a combined scaling factor of 75%/56.6%.

If you think the pawns above are too large, take a look at this picture from the 2016 Sinquefield Cup Tournament at the St. Louis chess club.
Your looking at a set with a 1.85" diameter king on a 2.17" (55mm) chess board, which gives you a scaling factor of a whopping 85%! And guess what? None of the participants complained. There were no protests. Nobody refused to accept their winnings because the dimensions of the chess sets didn't meet USCF regulations. Hey folks, these are not universal laws written on stone tablets, they're simply suggested guidelines. If your setup looks good and you like it and your opponent doesn't complain, then enjoy yourself and have fun. Which is, after all, what it's really all about.
Thank you very much, RussBell, for the links! Those are good sets, though I don't quite care for the knights on the second set. The first one has knights that I like, but I find them a little too big. I'm very fussy about my knights, it turns out: I don't like knights that are nearly the same height as a queen. This is just my personal preference.
I find it interesting that so many people think that 3.75" kings with base diameters of 1.6" are fine for 2" boards when the theoretically-ideal size would be 1.5" for a 2" square size board. But I think we are on the same page that 81% should be ok. Again, this is only a personal preference, but I would like a 3.5" king with a base diameter of 1.5" to be used on a 2" board. It's hard to find 3.5"'s with 1.5"'s.
Thanks for the tip, by the way, regarding Chess House. It's always good to find a company that takes customer service seriously.
loubalch, thanks again for your comments. I looked at my HOS Players pawns. I didn't measure them, but I put four of them on a square I drew that measured approx. 2.16" (or as close as I could get it) and they protrude only very slightly--like your picture #3 on the thread that you mentioned.
If I put two pawns diagonally inside a square on my 2" board (the one shown above with the blue squares), they do fit perfectly. This means that on a 2.16" squares board, they should fit even better, since I prefer a roomier look. I think, then, that I will try to order that 2.16" squares board. I'll post a few pictures of that board with the HOS Players pieces once I get it.
--
By the way, I loved your remarks regarding the picture from St. Louis! Yes, that is one tight fit, and it still looks ok (partly because the pieces recall the tighter fits of yesteryear). And I also agree with your general sentiments that these proportions are not commandments carved in stone that all people must follow at all times. One question, though: why are the Bishops not facing forwards? ;-)
Chess pieces and board question: I have a set with a king, 3.75" tall with a base of 1.75" in diameter. The set is best placed on a 2.25" squares board, I know. However, would things be too crowded if I put it on a board with 2.16" squares instead? Put another way: how important is that 0.09th of an inch? (I'm hoping the answer is "negligible," but I'd like to know what others think.) I have various practical reasons for asking this relating to how easily I can get a board where I am in Canada.